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Is there any way to deduce which invariance gives which conservation? I mean for example: how can you tell that time invariance is the one paired with conservat

Is there any way to deduce which invariance gives which conservation? I mean for example: how can you tell that time invariance is the one paired with conservation of energy? Why is e.g. time invariance not paired with momentum, current, or anything else, but specifically energy? I know that I can remember momentum is paired with translation simply because there's both the angular momentum and the non-angular momentum one and in space you have translation and rotation, so for time energy is the only one that's left over, but I'm not looking for a trick to remember it, I'm looking for the fundamental reason, as well as how to tell what will be paired with some invariance when looking at some other new invariance

chunky1994 3y ago
The conserved quantity is derived from Noether's theorem itself. One thing that is a bit hairy is that Noether's theorem only applies to a continuous, smooth (physical -> there is some wiggle room here) space. When deriving the conservation of energy from Noether's theorem you basically say that your Lagrangian (which is just a set of equations that describes a physical system) is invariant over time. When you do that you automatically get that energy is conserved. Each invariant produces a conserved quantity as explained in parent comment when you apple a specific transformation that is supposed to not change the system (i.e remain invariant). Now in doing this you're also invoking the principle of least action (by using Lagrangians to describe the state of a physical system) but that is a separate topic.
Cleonis 3y ago
In retrospect: the earliest recognition of a conserved quantity was Kepler's law of areas. Isaac Newton later showed that Kepler's law of areas is a specific instance of a property that obtains for any central force, not just the (inverse square) law of gravity. About symmetry under change of orientation: for a given (spherically symmetric) source of gravitational interaction the amount of gravitational force is the same in any orientation. For orbital motion the motion is in a plane, so for the case of orbital motion the relevant symmetry is cilindrical symmetry with respect to the plane of the orbit. The very first derivation that is presented in Newton's Principia is a derivation that shows that for any central force we have: in equal intervals of time equal amounts of area are swept out. (The swept out area is proportional to the angular momentum of the orbiting object. That is, the area law anticipated the principle of conservation of angular momentum) A discussion of Newton's derivation, illustrated with diagrams, is available on my website: http://cleonis.nl/physics/phys256/angular_momentum.php http://cleonis.nl/physics/phys256/angular_momentum.php The thrust of the derivation is that if the force that the motion is subject to is a central force (cilindrical symmetry) then angular momentum is conserved. So: In retrospect we see that Newton's demonstration of the area law is an instance of symmetry-and-conserved-quantity-relation being used. Symmetry of a force under change of orientation has as corresponding conserved quantity of the resulting (orbiting) motion: conservation of angular momentum. About conservation laws: The law of conservation of angular momentum and the law of conservation of momentum are about quantities that are associated with specific spatial characteristics, and the conserved quantity is conserved over time. I'm actually not sure about the reason(s) for classification of conservation of energy. My own view: we have that kinetic energy is not associated with any form of keeping track of orientation; the velocity vector is squared, and that squaring operation discards directional information. More generally, Energy is not associated with any spatial characteristic. Arguably Energy conservation is categorized as associated with symmetry under time translation because of absence of association with any spatial characteristic.
calhoun137 3y ago
The key point is that energy, momentum, and angular momentum are additive constants of the motion, and this additivity is a very important property that ultimately derives from the geometry of the space-time in which the motion takes place. > Is there any way to deduce which invariance gives which conservation? Yes. See Landau vol 1 chapter 2 [1]. > I'm looking for the fundamental reason, as well as how to tell what will be paired with some invariance when looking at some other new invariance I'm not sure there is such a "fundamental reason", since energy, momentum, and angular momentum are by definition the names we give to the conserved quantities associated with time, translation, and rotation. You are asking "how to tell what will be paired with some invariance" but this is not at all obvious in the case of conservation of charge, which is related to the fact that the results of measurements do not change when all the wavefunctions are shifted by a global phase factor (which in general can depend on position). I am not aware of any way to guess or understand which invariance is tied to which conserved quantity other than just calculating it out, at least not in a way that is intuitive to me. [1] https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%201%20-%20Landau%2C%20Lifshitz%20-%20Mechanics%20%283rd%20ed%2C%201976%29.pdf https://ia803206.us.archive.org/4/items/landau-and-lifshitz-...
- Aardwolf 3y ago
  But momentum is also conserved over time, as far as I know 'conservation' of all of these things always means over time. "In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant." That means it's conserved over time, right? So why is energy the one associated with time and not momentum?
  - calhoun137 3y ago
    my understandinf is that conservation of momentum does not mean momentum is conserved as time passes. it means if you have a (closed) system in a certain configuration (not in an external field) and compute the total momentum, the result is independent of the configuration of the system.

chunky1994 3y ago

The conserved quantity is derived from Noether's theorem itself. One thing that is a bit hairy is that Noether's theorem only applies to a continuous, smooth (physical -> there is some wiggle room here) space. When deriving the conservation of energy from Noether's theorem you basically say that your Lagrangian (which is just a set of equations that describes a physical system) is invariant over time. When you do that you automatically get that energy is conserved. Each invariant produces a conserved quantity as explained in parent comment when you apple a specific transformation that is supposed to not change the system (i.e remain invariant). Now in doing this you're also invoking the principle of least action (by using Lagrangians to describe the state of a physical system) but that is a separate topic.

Cleonis 3y ago

In retrospect: the earliest recognition of a conserved quantity was Kepler's law of areas. Isaac Newton later showed that Kepler's law of areas is a specific instance of a property that obtains for any central force, not just the (inverse square) law of gravity. About symmetry under change of orientation: for a given (spherically symmetric) source of gravitational interaction the amount of gravitational force is the same in any orientation. For orbital motion the motion is in a plane, so for the case of orbital motion the relevant symmetry is cilindrical symmetry with respect to the plane of the orbit. The very first derivation that is presented in Newton's Principia is a derivation that shows that for any central force we have: in equal intervals of time equal amounts of area are swept out. (The swept out area is proportional to the angular momentum of the orbiting object. That is, the area law anticipated the principle of conservation of angular momentum) A discussion of Newton's derivation, illustrated with diagrams, is available on my website: http://cleonis.nl/physics/phys256/angular_momentum.php http://cleonis.nl/physics/phys256/angular_momentum.php The thrust of the derivation is that if the force that the motion is subject to is a central force (cilindrical symmetry) then angular momentum is conserved. So: In retrospect we see that Newton's demonstration of the area law is an instance of symmetry-and-conserved-quantity-relation being used. Symmetry of a force under change of orientation has as corresponding conserved quantity of the resulting (orbiting) motion: conservation of angular momentum. About conservation laws: The law of conservation of angular momentum and the law of conservation of momentum are about quantities that are associated with specific spatial characteristics, and the conserved quantity is conserved over time. I'm actually not sure about the reason(s) for classification of conservation of energy. My own view: we have that kinetic energy is not associated with any form of keeping track of orientation; the velocity vector is squared, and that squaring operation discards directional information. More generally, Energy is not associated with any spatial characteristic. Arguably Energy conservation is categorized as associated with symmetry under time translation because of absence of association with any spatial characteristic.

calhoun137 3y ago

The key point is that energy, momentum, and angular momentum are additive constants of the motion, and this additivity is a very important property that ultimately derives from the geometry of the space-time in which the motion takes place. > Is there any way to deduce which invariance gives which conservation? Yes. See Landau vol 1 chapter 2 [1]. > I'm looking for the fundamental reason, as well as how to tell what will be paired with some invariance when looking at some other new invariance I'm not sure there is such a "fundamental reason", since energy, momentum, and angular momentum are by definition the names we give to the conserved quantities associated with time, translation, and rotation. You are asking "how to tell what will be paired with some invariance" but this is not at all obvious in the case of conservation of charge, which is related to the fact that the results of measurements do not change when all the wavefunctions are shifted by a global phase factor (which in general can depend on position). I am not aware of any way to guess or understand which invariance is tied to which conserved quantity other than just calculating it out, at least not in a way that is intuitive to me. [1] https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%201%20-%20Landau%2C%20Lifshitz%20-%20Mechanics%20%283rd%20ed%2C%201976%29.pdf https://ia803206.us.archive.org/4/items/landau-and-lifshitz-...

Aardwolf 3y ago
But momentum is also conserved over time, as far as I know 'conservation' of all of these things always means over time. "In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant." That means it's conserved over time, right? So why is energy the one associated with time and not momentum?
- calhoun137 3y ago
  my understandinf is that conservation of momentum does not mean momentum is conserved as time passes. it means if you have a (closed) system in a certain configuration (not in an external field) and compute the total momentum, the result is independent of the configuration of the system.